Uniform Recursive Tree Research Articles

Broadcasting on random recursive trees

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability 1−q and the opposite value with probability q, where q∈[0,1]. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from 1/2. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

Limit Theorems for the ‘Laziest’ Minimal Random Walk Model of Elephant Type

We consider a minimal model of one-dimensional discrete-time random walk with step-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): The walker can move forward (never backward), or remain at rest. For each $n=1,2,\cdots$, a random time $U_n$ between $1$ and $n$ is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time $U_n$, then at time $n+1$ it can move forward with probability $p$ [resp. $q$], or with probability $1-p$ [resp. $1-q$] it remains at its present position. For the case $q>0$, several limit theorems are obtained by Coletti, Gava, and de Lima (2019). In this paper we prove limit theorems for the case $q=0$, where the walker can exhibit all three forms of asymptotic behavior as $p$ is varied. As a byproduct, we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees.

Asymptotic results on Hoppe trees and their variations

AbstractA uniform recursive tree on n vertices is a random tree where each possible $(n-1)!$ labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.

The cut‐tree of large trees with small heights

AbstractWe destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017

From trees to seeds: On the inference of the seed from large trees in the uniform attachment model

We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global “balancedness” properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.

A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees

Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_{n}(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process \[D_{n}(u):=m^{\frac{1}{2}n}(W_{\infty}(\frac{u}{\sqrt{n}})-W_{n}(\frac{u}{\sqrt{n}}))\] converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result, we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Random Structures Algorithms 46 (2015) 346–361], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger’s theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that \[\mathcal{L}\{\sqrt{\frac{n}{2\log n}}(\operatorname{EPL}_{\infty}-\frac{\operatorname{EPL}_{n}-2n\log n}{n})\Big |\mathcal{G}_{n}\}\overset{\mathrm{a.s.w.}}{\underset{n\to\infty}\longrightarrow}\{\omega \mapsto\mathcal{N}_{0,1}\},\] where $\operatorname{EPL}_{n}$ is the external path length of a binary search tree $X_{n}$ with $n$ vertices, $\operatorname{EPL}_{\infty}$ is the limit of the Régnier martingale and $\mathcal{L}\{\cdot |\mathcal{G}_{n}\}$ denotes the conditional distribution w.r.t. the $\sigma$-algebra $\mathcal{G}_{n}$ generated by $X_{1},\ldots,X_{n}$. Almost sure weak convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton–Watson processes and the Pólya urn.

On leaf related statistics in recursive tree models

Using a bijection between a uniformly random permutation and a uniform recursive tree (URT), we give a simple proof of a recent result of Zhang that shows the asymptotic normality of the number of leaves in a URT with convergence rates. We also show that a similar result holds for a more general class of statistics related to URTs, that we call the number of runs of leaves in a URT. The second, and the main, purpose of the current note is to introduce and to study a non-uniform recursive tree model by exploiting the bijection between permutations and recursive trees. This may provide a useful framework for constructing various types of random trees.

Laplacian spectrum of a family of recursive trees and its applications in network coherence

Many of the topological and dynamical properties of a network are related to its Laplacian spectrum; these properties include network diameter, Kirchhoff index, and mean first-passage time. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. We choose a family of uniform recursive trees as our model, and propose a method to calculate the first- and second-order network coherence. Using the tree structures, we identify a relationship between the Laplacian matrix and Laplacian eigenvalues. We then derive the exact solutions for the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. We also obtain the scalings of network coherence with network size. The scalings of network coherence of the studied trees are smaller than those of Vicsek fractals and are not related to its fractal dimension.

A small-world network derived from the deterministic uniform recursive tree by line graph operation**Supported by The Ministry of Science and Technology 973 project (No. 2010C B334708); National Science Foundation of China (Nos. 61164005, 11161037, 11101232, 11461054, 11551001); The Ministry of education scholars and innovation team support plan of Yangtze River (No. IRT1068); Qinghai Province Nature Science Foundation Project (Nos. 2012-Z-943,

The deterministic uniform recursive tree is one of the deterministic versions of the uniform recursive tree . Zhang et al (2008 Eur. Phys. J. B 63 507–13) studied the properties of DURT, including its topological characteristics and spectral properties. Although DURT shows a logarithmic scaling with the size of the network, DURT is not a small-world network since its clustering coefficient is zero. Lu et al (2012 Physica A 391 87–92) proposed a deterministic small-world network by adding some edges with a simple rule in each DURT iteration. In this paper, we intoduce a method for constructing a new deterministic small-world network by the line graph operation in each DURT iteration. The line graph operation brings about cliques at each node of the previous given graph, and the resulting line graph possesses larger clustering coefficients. On the other hand, this operation can decrease the diameter at almost one, then giving the analytic solutions to several topological characteristics of the model proposed.

Fires on large recursive trees

We consider random dynamics on a uniform random recursive tree with n vertices. Successively, in a uniform random order, each edge is either set on fire with some probability pn or fireproof with probability 1−pn. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as n→∞, and prove a phase transition when pn is of order lnn/n. We then study the connectivity of the fireproof forest, more precisely the existence of a giant component. We finally investigate the sizes of the burnt subtrees.

Depth Properties of scaled attachment random recursive trees

AbstractWe study depth properties of a general class of random recursive trees where each node i attaches to the random node \documentclass{article} \usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty} \begin{document} \begin{align*}\left\lfloor iX_i\right\rfloor\end{align*} \end{document} and X0,…,Xn is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (sarrt). We prove that the typical depth Dn, the maximum depth (or height) Hn and the minimum depth Mn of a sarrt are asymptotically given by Dn ∼μ‐1 log n, Hn ∼ αmax log n and Mn ∼ αmin log n where μ,αmax and αmin are constants depending only on the distribution of X0 whenever X0 has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees Hn ∼ elog n that does not use branching random walks.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

A small-world network derived from the deterministic uniform recursive tree

As the deterministic version of the uniform recursive tree (URT), the deterministic uniform recursive tree (DURT) has been intensively studied by Zhang et al. (2008) [21]. They gave several important properties of DURT, including its topological characteristics and spectral properties. Although DURT shows a logarithmic scaling with the size of the network, DURT is not a small-world network since its clustering coefficient is zero. In this paper, we propose a new deterministic small-world network by adding some edges with a simple rule in each DURT iteration, and then give the analytic solutions to several topological characteristics of the model proposed.

Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees

The determination of mean first-passage time (MFPT) for random walks in networks is a theoretical challenge, and is a topic of considerable recent interest within the physics community. In this paper, according to the known connections between MFPT, effective resistance, and the eigenvalues of graph Laplacian, we first study analytically the MFPT between all node pairs of a class of growing treelike networks, which we term deterministic uniform recursive trees (DURTs), since one of its particular cases is a deterministic version of the famous uniform recursive tree. The interesting quantity is determined exactly through the recursive relation of the Laplacian spectra obtained from the special construction of DURTs. The analytical result shows that the MFPT between all couples of nodes in DURTs varies as N ln N for large networks with node number N. Second, we study trapping on a particular network of DURTs, focusing on a special case with the immobile trap positioned at a node having largest degree. We determine exactly the average trapping time (ATT) that is defined as the average of FPT from all nodes to the trap. In contrast to the scaling of the MFPT, the leading behavior of ATT is a linear function of N. Interestingly, we show that the behavior for ATT of the trapping problem is related to the trapping location, which is in comparison with the phenomenon of trapping on fractal T-graph although both networks exhibit tree structure. Finally, we believe that the methods could open the way to exactly calculate the MFPT and ATT in a wide range of deterministic media.

Weight of a link in a shortest path tree and the Dedekind Eta function

AbstractThe weight of a randomly chosen link in the shortest path tree on the complete graph with exponential i.i.d. link weights is studied. The corresponding exact probability generating function and the asymptotic law are derived. As a remarkable coincidence, this asymptotic law is precisely the same as the distribution of the cost of one “job” in the random assignment problem. We also show that the asymptotic (scaled) maximum interattachment time to that shortest path tree, which is a uniform recursive tree, equals the square of the Dedekind Eta function, a central function in modular forms, elliptic functions, and the theory of partitions. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

The height of scaled attachment random recursive trees

We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.

Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

As one of the most significant models, the uniform recursive tree (URT) has found many applications in a variety of fields. In this paper, we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees (DURTs) that are deterministic versions of URT. Firstly, from the perspective of complex networks, we investigate analytically the main structural characteristics of DURTs, and obtain the accurate solutions for these properties, which include degree distribution, average path length, distribution of node betweenness and degree correlations. Then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for DURTs. Our research may shed light on a better understanding of the features for URT. Also, the analytical methods used here are capable of being extended to many other deterministic networks, making the precise computation of their properties (especially the full spectrum characteristics) possible.

Topologies and Laplacian spectra of a deterministic uniform recursive tree

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.

Degree and component size distributions in the generalized uniform recursive tree

We propose a generalized model for the famous uniform recursive tree (URT) by introducing an imperfect growth process, which may generate disconnected components (clusters). The model undergoes an interesting phase transition from a singly connected network to a graph consisting of fully isolated nodes. We investigate the distributions of degree and component sizes by both theoretical predictions and numerical simulations. For the nontrivial cases, we show that the network has an exponential degree distribution while its component size distribution follows a power law, both of which are related to the imperfect growth process. We also predict the growth dynamics of the individual components. All analytical solutions are successfully contrasted with computer simulations.

Limiting behavior of uniform recursive trees

The authors consider the limiting behavior of various branches in a uniform recursive tree with size growing to infinity. The limiting distribution of ζ n,m, the number of branches with size m in a uniform recursive tree of order n, converges weakly to a Poisson distribution with parameter 1 m with convergence of all moments. The size of any large branch tends to infinity almost surely.